Given a = 31.5, b = 51.8, A = 33 degrees, find the value of c in triangle ABC.

bsinA = 51.8sin33 = 28.21 -> bsinA < a < b so there are two solutions

Using sine law:

sinB/b = sinA/a
sinB/51.8 = sin33/31.5
sinB = 51.8 * sin33/31.5
sinB = 0.895628635
B = arcsin(0.895628635)
B = 63.6

C = 180 - 33 - 63.6 = 83.4

c/sinC = a/sinA
c/sin83.4 = 31.5/sin33
c = sin83.4 * 31.5/sin33
c = 57

The second solution is 29, but not sure how to solve for it.

there are two angles whose sine is 0.895...

Quad I and Quad II

To solve for the second solution, you can use the following steps:

1. Since we already found angle B to be 63.6 degrees, we can use the sine law again to find the second solution for c.
2. We know that sinC/sinB = c/b
Let's denote the second solution for c as c2 and b as b2.
So, sinC/sin63.6 = c2/b2
3. We can rearrange this equation to solve for c2:
c2 = sinC * b2 / sin63.6
4. From the previous steps, we found that angle C is 83.4 degrees and angle B is 63.6 degrees.
Let's assume that angle A is still 33 degrees, then angle C + angle A + angle B should equal 180 degrees. We can use this information to find angle C using this equation:
180 = 33 + 63.6 + C
C = 83.4 degrees
5. Now, let's substitute the values we know into the equation for c2:
c2 = sin83.4 * b2 / sin63.6
6. The value of b2 is given as 51.8, so substituting this value into the equation gives:
c2 = sin83.4 * 51.8 / sin63.6
7. Evaluating this expression gives the value of c2:
c2 = 109

Therefore, the two solutions for the value of c in triangle ABC are c = 57 and c = 109.

To solve for the second solution, we can use the property of the sine function that states sin(x) = sin(180 - x). In other words, the sine of an angle is equal to the sine of its supplementary angle.

In this case, we have:

sin(B) = sin(180 - B)

So, we can calculate B for the second solution as follows:

B = 180 - arcsin(0.895628635)
B = 116.4

Now, the angle C for the second solution can be calculated using:

C = 180 - A - B
C = 180 - 33 - 116.4
C = 30.6

To find the side c for the second solution, we use the sine law:

c/sinC = a/sinA
c/sin30.6 = 31.5/sin33

Rearranging this equation to solve for c:

c = sin30.6 * 31.5/sin33
c = 28.75

Therefore, the two solutions for the value of c in triangle ABC are approximately 57 and 28.75.